F01 Chapter Contents
F01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF01ABF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F01ABF calculates the accurate inverse of a real symmetric positive definite matrix, using a Cholesky factorization and iterative refinement.

## 2  Specification

 SUBROUTINE F01ABF ( A, LDA, N, B, LDB, Z, IFAIL)
 INTEGER LDA, N, LDB, IFAIL REAL (KIND=nag_wp) A(LDA,N), B(LDB,N), Z(N)

## 3  Description

To compute the inverse $X$ of a real symmetric positive definite matrix $A$, F01ABF first computes a Cholesky factorization of $A$ as $A=L{L}^{\mathrm{T}}$, where $L$ is lower triangular. An approximation to $X$ is found by computing ${L}^{-1}$ and then the product ${L}^{-\mathrm{T}}{L}^{-1}$. The residual matrix $R=I-AX$ is calculated using additional precision, and a correction $D$ to $X$ is found by solving $L{L}^{\mathrm{T}}D=R$. $X$ is replaced by $X+D$, and this iterative refinement of the inverse is repeated until full machine accuracy has been obtained.
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag

## 5  Parameters

1:     A(LDA,N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the upper triangle of the $n$ by $n$ positive definite symmetric matrix $A$. The elements of the array below the diagonal need not be set.
On exit: the lower triangle of the inverse matrix $X$ is stored in the elements of the array below the diagonal, in rows $2$ to $n+1$; ${x}_{ij}$ is stored in ${\mathbf{A}}\left(i+1,j\right)$ for $i\ge j$. The upper triangle of the original matrix is unchanged.
2:     LDA – INTEGERInput
On entry: the first dimension of the array A as declared in the (sub)program from which F01ABF is called.
Constraint: ${\mathbf{LDA}}\ge {\mathbf{N}}+1$.
3:     N – INTEGERInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 1$.
4:     B(LDB,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the lower triangle of the inverse matrix $X$, with ${x}_{ij}$ stored in ${\mathbf{B}}\left(i,j\right)$, for $i\ge j$.
5:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F01ABF is called.
Constraint: ${\mathbf{LDB}}\ge {\mathbf{N}}$.
6:     Z(N) – REAL (KIND=nag_wp) arrayWorkspace
7:     IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
The matrix $A$ is not positive definite, possibly due to rounding errors.
${\mathbf{IFAIL}}=2$
The refinement process fails to converge, i.e., the matrix $A$ is ill-conditioned.
${\mathbf{IFAIL}}=3$
${\mathbf{N}}<1$, or ${\mathbf{LDA}}<{\mathbf{N}}+1$, or ${\mathbf{LDB}}<{\mathbf{N}}$.

## 7  Accuracy

The computed inverse should be correct to full machine accuracy. For a detailed error analysis see page 40 of Wilkinson and Reinsch (1971).

The time taken by F01ABF is approximately proportional to ${n}^{3}$.

## 9  Example

This example finds the inverse of the $4$ by $4$ matrix:
 $5 7 6 5 7 10 8 7 6 8 10 9 5 7 9 10 .$

### 9.1  Program Text

Program Text (f01abfe.f90)

### 9.2  Program Data

Program Data (f01abfe.d)

### 9.3  Program Results

Program Results (f01abfe.r)